Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 295, whose code is 4, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1248, Form 0 \( \not \Rightarrow \) Form 293 whose summary information is:
    Hypothesis Statement
    Form 0  \(0 = 0\).

    Conclusion Statement
    Form 293 <p> For all sets \(x\) and \(y\), if \(x\) can be linearly ordered and there is a mapping of \(x\) onto \(y\), then \(y\) can be linearly ordered. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 2010, whose string of implications is:
    295 \(\Rightarrow\) 30 \(\Rightarrow\) 293

The conclusion Form 0 \( \not \Rightarrow \) Form 295 then follows.

Finally, the
List of models where hypothesis is true and the conclusion is false:

Name Statement
\(\cal N28\) Blass' Permutation Model The set \(A=\{a^i_{\xi}: i\in \Bbb Z, \xi\in\aleph_1\}\)
\(\cal N37\) A variation of Blass' model, \(\cal N28\) Let \(A=\{a_{i,j}:i\in\omega, j\in\Bbb Z\}\)

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